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We consider the problem of estimating the expected time to find a maximumdegree node on a graph using a (parameterized) biased random walk. Forassortative graphs the positive degree correlation serves as a local gradientfor which a bias towards selecting higher degree neighbors will on averagereduce the search time. Unfortunately, although the expected absorption time onthe graph can be written down using the theory of absorbing Markov chains,computing this time is infeasible for large graphs. With this motivation, weconstruct an absorbing Markov chain with a state for each degree of the graph,and observe computing the expected absorption time is now computationallyfeasible. Our paper finds preliminary results along the following lines: i)there are graphs for which the proposed Markov model does and graphs for whichthe model does not capture the absorbtion time, ii) there are graphs whererandom sampling outperforms biased random walks, and graphs where biased randomwalks are superior, and iii) the optimal bias parameter for the random walk isgraph dependent, and we study the dependence on the graph assortativity.

A Markov chain model for the search time for max degree nodes in a graph using a biased random walk